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4Circles.notebook
1
November20,2013
O
DONOW:1.Whatarecoordinatesofthecenterofthecircle?
2.Whatisthelengthoftheradius?
CIRCLES
(h,k)r
The equation of a circle:
(x-h)2 + (y- k)2 = r2
where (h,k) represents the CENTER
and r represents the RADIUS
To find the center and radius when you are given the
equation of a circle:
-switch the signs for the coordinates of the center
-take the square root of "r2" to get the radius
For each of the following circles, find the center and the radius:
1) x2 + y2 = 25 Center: __________ Radius: ___________
2) (x - 4)2 + (y + 8)2 = 4 Center: __________ Radius: ____________
3) (x + 9)2 + (y - 1)2 = 10 Center: __________ Radius: ____________
4) (x - 2)2 + y2 = 5 Center: __________ Radius: ____________
Given the following, write an equation for each circle.
5) Center: (-3,2) Radius: 6
6) Center: (0, -8) Radius: 1
7) Center: (-7,-10)Radius: 9
8) Write an equation of a circle whose center is
(3, -5) and whose diameter is 10.
Writing the Equation of a Circle Given a Graph:
9) Write an equation for circle O shown below.
10) Write an equation for the circle shown below.
Circles: VOCABULARY
4Circles.notebook
2
November20,2013
Example3
Example5
APandBPtouchthecircle___________
4Circles.notebook
3
November20,2013
EXAMPLES1) Given: CE = 12, ED = 12, and AE = 3, find BE.
2) In the diagram of circle O below, chord AB intersects
chord CD at E, DE = 2x + 8, EC = 3, AE = 4x - 3, and
EB = 4. What is the value of x?
3) Chords AB and CD intersect at point E in a circle with
center at 0. If AE = 8, AB = 20, and DE = 16, what is the
length of CE? EXTERNAL times WHOLE secant = EXTERNAL times WHOLE secant
E D
C
AB
ew=ew
External segment of a secant --> part of secant segment that is
outside the circle.
Intersecting Secants
4) In the accompanying diagram, PAB and PCD are
secants drawn to circle O, PA = 8, PB = 20, and PD = 16.
What is the length of PC?
5) Secants JKL and JMN are drawn to circle O from an
external point, J. If JK = 8, LK = 4, and JM = 6, what is
the length of JN?
4Circles.notebook
4
November20,2013
6) In the diagram below of circle O, secant AB intersects
circle O at D, secant AOC intersects circle O at E, AE = 4,
AB = 12, and DB = 6. What is the length of OC? EXTERNAL times WHOLE secant = TANGENT times TANGENT
A
PO
BC
ew=tt
Tangent Intersecting a Secant
7) In the diagram below, tangent PA and secant PBC are
drawn to circle O from external point P. If PB = 4 and
BC = 5, what is the length of PA?
8) In the diagram below, tangent AB and secant ACD are
drawn to circle O from an external point A, AB = 8 and
AC = 4. What is the length of CD?
9) Secant ABC and tangent AD are drawn to circle O from
external point A. If AD = 10 and AC = 20, find the length
of AB.
DO NOW:Given: AB = 7, BC = 8, and AD = 3, find DE.
4Circles.notebook
5
November20,2013
Directions: Use the appropriate segment relationship to solve for
the missing measurement. If necessary, specify your answer in
simplest radical form.
1) Find x in the circle below.
2) Find x and the measure of ED in the circle below.
3) Given: AD = x, DE = 2, AB = 2, and CB = 10, find AD
and AE.
4) In circle O below, AE = 6, EB = 4, and CE = ED. Find CE
in simplest radical form.
A
B C
D
EO
5) In the circle below, find y.Then, find x to the nearest tenth.
6) In the diagram below of circle O, chords RT and QS intersect at M.
Secant PTR and tangent PS are drawn to circle O. The length of RM is
two more than the length of TM, QM = 2, SM = 12, and PT = 8. Find the
length of RT and PS.
4Circles.notebook
6
November20,2013
DO NOW:
A segment that touches a circle ONLY one time is
called a ____________________.
Fill in the blank below:
Theorems Involving Tangents To Circles
Common tangents: lines or segments that are tangent to more than one
circle at the same time
Tangents From The Same External Point-tangents to a circle from the same external point are equal.
(You may think of this as the "Hat"
Theorem because the diagram looks
like a circle wearing a pointed hat.)
Which two segments in
the circle above are equal?
Examples:1) What is the value of x in the circle below?
2) In the diagram below, circles X and Y have two tangents
drawn to them from external point T. The points of tangency are
C, A, S, and E. The ratio of TA to AC is 1:3. If TS = 24, find the
length of SE.
x28x
20
3) What is the value of x in the circle below?
What is the length of AB?
4) In the diagram below, triangle ABC is circumscribed about circle O
and the sides of triangle ABC are tangent to the circle at points D, E,
and F. If AB = 20, AE = 12, and CF = 12, what is the length of AC?
4Circles.notebook
7
November20,2013
5) In the diagram below, AB, BC, and AC are tangents to
circle O at points F, E, and D, respectively, AF = 6, CD = 5,
and BE = 4. What is the perimeter of triangle ABC?
Tangents Drawn To The Radius Of A Circle
If a line is tangent to a circle, it is perpendicular to the radius drawn
to the point of tangency.
If we connect points O and B, what type of triangle is formed?
Examples:
1) In the diagram below of triangle PAO, AP is tangent to
circle O at point A, OB = 7, and BP = 18. What is the
length of AP?
2) In the diagram below, AC and AD are tangent to circle B at
points C and D respectively, and BC, BD, and BA are drawn. If
AC = 12 and AB = 15, what is the length of BD?
DO NOW:
O
A
BC
D
In the diagram below, AB and CD are ________________.
OA and OB are ___________________.
Theorems About Chords and Circles-a radius perpendicular to a chord bisects the chord. (cuts chord in half)
-a radius that bisects a chord is perpendicular to the chord.
-the perpendicular bisector of a chord passes through the center of the circle.
If we connect points O and A we can
make a _____________________________.
4Circles.notebook
8
November20,2013
Examples:1) In the diagram below of circle O, diameter AOB is
perpendicular to chord CD at point E, OA = 6, and OE =2. What
is the length of CE, in simplest radical form?
2) In circle O shown below, diameter DB is perpendicular to chord
AC at E. If DB = 34, AC = 30, and DE > BE, what is the length of
BE?
3) In the diagram below of circle O, radius OC is 5cm. Chord AB is
8cm and is perpendicular to OC at point P. What is the length of
OP, in centimeters?
4) In circle O shown below, chords AB and CD and radius OA are
drawn, such that AB = CD, OE is perpendicular to AB, OF is
perpendicular to CD, OF = 16, CF = y + 10, and CD = 4y - 20.
Find the length of DF.
Determine the length of OA.
Another Theorem About Chords:
-the closer a chord is to the center of a circle, the
longer the chord.
5) In the diagram below, triangle ABC is inscribed in circle P. The
distances from the center of circle P to each side of the triangle are
shown.
Which statement about the sides of the triangle is true?
1) AB > AC > BC
2) AB < AC and AC > BC
3) AC > AB > BC
4) AC = AB and AB > BC
DO NOW:When lines are a constant distance apart from each
other, as shown in the diagram below, they are called
_______________________.
4Circles.notebook
9
November20,2013
Parallel ChordsIn a circle, parallel chords intercept congruent(equal) arcs.
Examples:1) In the diagram of circle O below, chord CD is parallel
to diameter AOB and mAC = 30. What is mCD?
2) In the diagram below of circle O, diameter AB is
parallel to chord CD. If mCD = 70, what is mAC?3) In the diagram of circle O below, chord CD is parallel to
diameter AOB and mCD = 110. What is mDB?
4) In circle O shown in the diagram below, chords AB and
CD are parallel. If mAB = 104 and mCD = 168, what is
mBD?
5) In the diagram below of circle O, chord AB is parallel to
chord CD. Which statement must be true?
is the symbol for CONGRUENT
4Circles.notebook
10
November20,2013
6) In the diagram below of circle O, chord AB ll chord CD,
and chord CD ll chord EF. Which statement must be true?
ll is the symbol for PARALLEL
7) In the diagram below, two parallel lines intersect
circle O at points A, B, C, and D with mAB = x + 20
and mDC = 2x - 20. Find x and mAB.
DONOW:
1.Nameacentralangle.
2.Nameaninscribedangle.
*USEYOURVOCABSHEET!*
CentralAngle:Acentralangleisanangleformedbytwointersectingradiisuchthatitsvertexisatthecenterofthecircle.
InscribedAngle:Aninscribedangleisananglewithitsvertex"on"thecircle,formedbytwointersectingchords.
Specialsituationsinvolvinginscribedangles:
Anangleinscribedinasemicircleisarightangle
Theoppositeanglesinacyclicquadrilateralaresupplementary.
Aquadrilateralinscribedinacircleiscalledacyclicquadrilateral.
4Circles.notebook
11
November20,2013
Examples:1) What is the value of x in the circle below?
2) Use what you know about central and inscribed angles
to find x in the circle below.
3) If the m
4Circles.notebook
12
November20,2013
Examples:
1) In the diagram below of circle O, chords AD and BC
intersect at E, the measure of arcAC = 87 and the measure of
arcBD = 35. What is the degree measure of
4Circles.notebook
13
November20,2013
7) In the accompanying diagram, AB is a diameter of
circle O and chord CD intersects diameter AB at E. If the
measure of arcAD = 100 and measure of arcAC = 40, find
m
4Circles.notebook
14
November20,2013
4) In the accompanying diagram of circle O, diameter AOB is extended
through B to external point P, tangent PC is drawn to point C on the
circle and the measure of arc AC : measure of arc BC = 7:2. Find
m
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Oct 25-2:41 PM DO NOW: Find the center and radius of each circle. 1) (x+ 8) 2 + y 2 = 4 2) x 2 + (y - 1) 2 = 6
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